#72321789

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#32554
commutator, Frattini, cosocle	C₂³×C₈	C₂⁴
prime radical	C₂³×D₈	C₂³
LCS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂×#2961 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1224 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 36 points:

(1,2)(3,4)(5,6,12,18,30,16,32,7,13,8,14,21,10,19,11,17)(9,20,26,36,24,35,27,22,29,23,31,33,28,34,25,15);(2,3)(5,7,10,21,13,17,30,18)(6,14,16,32,8,12,19,11)(9,23,24,35,29,20,28,34)(15,27,33,31,22,25,36,26);(1,3,4,2)(5,8,13,6)(7,14,17,12)(9,22)(10,19,30,16)(11,21,32,18)(15,29)(20,27)(23,25)(24,36)(26,35)(28,33)(31,34);(2,3)(5,9,10,24,13,29,30,28)(6,15,19,36,8,22,16,33)(7,20,18,35,17,23,21,34)(11,31,14,25,32,26,12,27)

Polycyclic presentation:

2^10|a^2=g,b^2=e,c^2=fhi,d^2=ghi,e^2=j,f^2=i,g^2=ij,h^2=i,i^2=j,b^a=bf,c^a=cfghi,c^b=cfhi,d^a=defj,d^b=dgh,d^c=defgij,e^a=ej,f^b=fi,f^c=fij,f^d=fi,g^b=gij,g^c=gi,g^d=gij,h^b=hi,h^c=hij,h^d=hij,i^b=ij,i^c=ij,i^d=ij

Conjugacy classes & character table

70 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72321841, #72322333
maximal normal subgroup of	#72321841, #72322333

GAP

Past the SmallGroups library coverage — no SmallGroups id. Construct it from its polycyclic presentation:

gap> F := FreeGroup(IsSyllableWordsFamily, "a", "b", "c", "d", "e", "f", "g", "h", "i", "j");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 2, 2, 2, 2, 2, 2]);;
gap> SetPower(coll, 1, F.7);;
gap> SetPower(coll, 2, F.5);;
gap> SetPower(coll, 3, F.6*F.8*F.9);;
gap> SetPower(coll, 4, F.7*F.8*F.9);;
gap> SetPower(coll, 5, F.10);;
gap> SetPower(coll, 6, F.9);;
gap> SetPower(coll, 7, F.9*F.10);;
gap> SetPower(coll, 8, F.9);;
gap> SetPower(coll, 9, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.6);;
gap> SetConjugate(coll, 3, 1, F.3*F.6*F.7*F.8*F.9);;
gap> SetConjugate(coll, 3, 2, F.3*F.6*F.8*F.9);;
gap> SetConjugate(coll, 4, 1, F.4*F.5*F.6*F.10);;
gap> SetConjugate(coll, 4, 2, F.4*F.7*F.8);;
gap> SetConjugate(coll, 4, 3, F.4*F.5*F.6*F.7*F.9*F.10);;
gap> SetConjugate(coll, 5, 1, F.5*F.10);;
gap> SetConjugate(coll, 6, 2, F.6*F.9);;
gap> SetConjugate(coll, 6, 3, F.6*F.9*F.10);;
gap> SetConjugate(coll, 6, 4, F.6*F.9);;
gap> SetConjugate(coll, 7, 2, F.7*F.9*F.10);;
gap> SetConjugate(coll, 7, 3, F.7*F.9);;
gap> SetConjugate(coll, 7, 4, F.7*F.9*F.10);;
gap> SetConjugate(coll, 8, 2, F.8*F.9);;
gap> SetConjugate(coll, 8, 3, F.8*F.9*F.10);;
gap> SetConjugate(coll, 8, 4, F.8*F.9*F.10);;
gap> SetConjugate(coll, 9, 2, F.9*F.10);;
gap> SetConjugate(coll, 9, 3, F.9*F.10);;
gap> SetConjugate(coll, 9, 4, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);